So what topics are you interested in?

[hubris]

I am looking forward to either i) knot theory or ii) algebraic geometry.

What about everyone else?

[origin415]

I haven't had courses in any of these, so I'm mainly going off what I've heard and the first few chapters of some introductory texts I've picked up, but I think algebraic geometry, differential geometry, or algebraic topology. I've chosen to apply to schools that seem to have a balance of each so that I can decide later, or even decide on something entirely different.

Edited January 14, 2010 by origin415

[Tam]

I put algebraic geometry, algebra, or combinatorics on most of my statements of purpose. I'll need to know more about those areas (and others) to decide, though. I feel like I just barely have a glimpse of the overall landscape of math.

[George]

My background is in applied mathematics, although my interests span pretty much everything I ever come across except probability and statistics. It's very unfortunate that the newest trend is for math programs to split into pure and applied tracks because I really wanted to get into grad school first and then find out which one I would like to pursue. The way it is, I was able to find at least a few schools where this schism hasn't occurred yet, and for the rest I grudgingly filled out the app for the applied track.

What bothers me is that I don't like the word "applied." If the application of my work happens to be in algebra or string theory, which I find hard to distinguish from algebra, then how in the world is it different from "pure" mathematics? A professor described the difference to me like so: "If you like proving theorems, do pure. If not, do applied." That makes sense to an extent, but it doesn't help me because I still like both. Sigh.

[origin415]

What bothers me is that I don't like the word "applied." If the application of my work happens to be in algebra or string theory, which I find hard to distinguish from algebra, then how in the world is it different from "pure" mathematics? A professor described the difference to me like so: "If you like proving theorems, do pure. If not, do applied." That makes sense to an extent, but it doesn't help me because I still like both. Sigh.

The difference isn't about theorems, I think. Rather, its motivation: do you wish to do math for its own sake because you find it intriguing and elegant, or do you wish to do math because it can help people, solve real problems, and better our understanding of our universe (as opposed to idealized, abstract mathematical universes).

Personally I'm completely 100% in the former category, I could care less about the ramifications, but think the ideas themselves are beautiful.

I think most people in mathematics studying string theory, by the way, would put themselves as pure.

The University of Washington would seem to agree with me, here is what they say about the fact that they have separate departments for mathematics and applied mathematics, from their FAQ:

"The most important difference, perhaps, is that faculty members in the two departments are interested in a somewhat different range of subjects. A second important difference is one of motivation: Research in the Applied Math Department is a bit more oriented toward using mathematical ideas to solve problems that arise outside of mathematics, while research in the Math Department is directed more at understanding the underlying mathematical ideas, whether motivated by applications or by more theoretical considerations."

[appmatharmy]

Pure vs. Applied

I would say that applied means your solving someone else's problem using mathematical tools.

While pure is doing it for its own sake (elegance, growth of knowledge, etc) and the means to expand that knowledge is by proving theorems, ergo prove a theorem to get a PhD.

If you have no interest in probability or stats I would lean toward pure, unless you have a knack for dynamical systems or Diff Eq.

[kdilks]

"Applied Math" just isn't a very well defined thing. The stuff that I do (algebraic combinatorics) is considered applied math in some places and pure math in others, even though it meets pretty much none of the criterion anybody has listed so far for being "applied math".

[kroner]

Yeah, a lot of times it has much more to do with whether a certain branch has historically been thought of as "applied" than whether the actual work in question has an application. I like algebraic geometry and topology, but I'm also interested in its applications to physics. That kind of stuff is classified as pure math at all the schools I'm applying to, regardless of whether it's being applied.

[George]

Yeah, a lot of times it has much more to do with whether a certain branch has historically been thought of as "applied" than whether the actual work in question has an application. I like algebraic geometry and topology, but I'm also interested in its applications to physics. That kind of stuff is classified as pure math at all the schools I'm applying to, regardless of whether it's being applied.

That's really cool. In another life, I would like to do mathematical physics. String theory, quantum field theory and general relativity are amazing feats of human intelligence.

Do you think it's possible for me to learn these things and do research in them in grad school if my training in physics is limited to 2 introductory level courses in college? They covered mechanics, electromagnetism, optics... you know, the basics. Did you double major in math and physics?

[kroner]

It's funny you should ask that.

When I started undergrad I was really into both math and physics, but I kinda got fed up with physics at my school and just focused on math. I only took 3 terms of it, which covered mechanics and EM, and also some special relativity. I graduated in 2008, and then took some time to figure out what I wanted to do with my life. I regretted not ever taking quantum mechanics, so I got a book and started studying and it was great. So great that for a while I decided I wanted to switch over and go to school for physics. I took the physics GRE, and started studying general relativity, and looking at what schools I wanted to apply to. But as I thought more about it I came to the conclusion that I still love math, and what really interests me about physics is the math end, which tends to be in math departments. Like if I'm going to study topology for physics, it's still topology. So I applied to schools for math, but to places where hopefully I'll have some options. I plan to keep studying both between now and the fall. I don't really know where it'll take me.

So to answer the question, I think absolutely if you are interested in physics and have the math background you could get into that stuff. I haven't done any QFT yet or string theory, but of what I've seen, if you have the math down already the physics is not too hard. Of course this is coming from someone who isn't actually doing any physics related research yet, or even been accepted to any schools yet for that matter, so you might want to take that advice with a grain of salt.

In fact it could be that I'm totally deluding myself.

Edited February 1, 2010 by kroner

[mathgrad]

I'm into commutative algebra and maybe algebraic geometry. I'm not really sure yet, though.

[HermanHesse216]

Inverse Problems.

And if I may add my favorite quote... "Pure Math is but a sub field of Applied Math."

[hmmmmm]

Wow! So many maybe-algebraic geometers here What preparation have you done for this?

[someDay]

Not that I'm into algebraic geometry, but to everyone interested in it I recommend the wonderful book "An invitation to algebraic geometry" by someone whose name I have forgotten (Karen Smith?), published by springer and available at ridiculous prices at your favourite bookshop.

Personally I'm interested in homological algebra and low-dimensional topology.. though this is subject to constant change.

sD.

[Slarti]

I want to do combinatorics, but I have no idea if that will end up being graph theory, probabilistic combinatorics, algebraic combinatorics, or some combination of those. I tried to apply to places with a wide variety of combinatorialists, but that turns out to be kinda difficult.

I'm surprised by the number of algebraic geometers, but then again I've somehow not taken any topology or geometry classes.

[kdilks]

I want to do combinatorics, but I have no idea if that will end up being graph theory, probabilistic combinatorics, algebraic combinatorics, or some combination of those. I tried to apply to places with a wide variety of combinatorialists, but that turns out to be kinda difficult.

Got any early favorites between Minnesota, Rutgers, and UIUC?

[Reynolds]

I loved both Real and Complex analysis as an undergraduate. I don't know where exactly that will lead me since it seems most people who research analysis use it to work on PDEs but I definitely want to explore each of those areas more.

I also really, really loved set theory and I'd be thrilled to research some kind of logic/set theory stuff. Pretty much I want to learn more about everything, but those are the areas I'm particularly excited about at the moment.

[Multiple Infusions]

I like math, particularly commutative algebra and algebraic geometry (like so many others in this forum :]). I'm yet to have a firm grasp on scheme theory, but I've really enjoyed what I've seen in classical algebraic geometry (the contravariance sometimes gets a little screwy though). I guess that is just one of many things to conquer in the next year.

[BongRips69]

Geometry, fluids, dynamical systems, modelling.

Also have more than a passing interest in logic. That's mainly because I'm a Platonist and view all mathematics/language/art as the quest to model some unchangeable truth. Sort of like THE BOOK that Erdos spoke of.

Edited March 25, 2010 by BongRips69

[Ostinattos]

Differential Geometry, Riemmanian, Symplectic, some Algebraic Topology and a little Geometric Analysis.

I took a course in Category Theory about a year ago and it was really good, except for the lack of bibliography, but I think of it as a good tool to have at my disposal rather than a subjet I would like to do research in.

[kaz]

I'm interested in mostly modelling, maybe climate science, nanotechnology, materials modelling etc.

Do most people do some research(like summer research or work with a prof etc) before deciding what they're interested in, or just try to decide base on what interests you during undergrad?

Edited September 1, 2010 by kaz

[mathgeek]

I really wish I had the chance to take algebraic geometry as an undergraduate, but there is no course in the subject at my university (we have one algebraic geometer, and he has given it as a grad course before).

My strongest interest is in general topology, but this isn't really an active research area anymore. My other main interests are set theory, and applications of set theory to other areas (especially topology and analysis). I'm hoping to end up doing some kind of set-theoretic topology, but it depends which grad schools accept me. Topological groups, functional analysis, as well as mathematical logic, are also topics I find increasingly interesting.

[hubris]

I really wish I had the chance to take algebraic geometry as an undergraduate, but there is no course in the subject at my university (we have one algebraic geometer, and he has given it as a grad course before).

My strongest interest is in general topology, but this isn't really an active research area anymore. My other main interests are set theory, and applications of set theory to other areas (especially topology and analysis). I'm hoping to end up doing some kind of set-theoretic topology, but it depends which grad schools accept me. Topological groups, functional analysis, as well as mathematical logic, are also topics I find increasingly interesting.

Looks like you have a healthy range of interests and you can always question your sanity and take AG later.